278 Prof. Smith on Complex Binary Quadratic Forms. [June 16, 



the enamel, flat and smooth to the stumps, exposing there a central tract 

 of osteodentine without any sign of decay. 



The paper is illustrated by a view and plans of the cavern, and by 

 figures of the principal human remains, and of two implements of bone on 

 which the Vicomte de Lastic had discovered, on removal of the breccia, 

 outline figures of the head of a reindeer and the head of a horse in profile. 



The description of the various remains of the animals killed for food, 

 and of the flint- and bone-implements apphed to that and other purposes, 

 will be the subject of a future communication. 



June 16, 1864. 



Major-General SABINE, President, in the Chair. 



Dr. Briuton; Professor Boole; Mr. T. Grubb ; Sir Charles Locoek,. 

 Bart. ; and Mr. Nicholas Wood, were admitted into the Society. 



The following communications were read : — 



I. "On Complex Binary Quadratic Forms.'' By H. J. Stephejj 

 Smith, M.A., F.B.S., Savilian Professor of Geometry in the 

 University of Oxford. Received May 18, 1864. 



The purpose of this note is to extend to complex quadratic forms some 

 important investigations of Gauss relating to real quadratic forms. We 

 shall consider in order (I.) the definition of the Genera, (II.) the theory 

 of Composition, (III.) the determination of the number of Ambiguous 

 Classes, (IV.) the representation of forms of the principal genus by 

 ternary quadratic forms of determinant 1. For the comparison of the 

 numbers of classes of different orders, we may refer to a paper by 

 M. Lipschitz (Crelle's Journal, vol. liv. p. 193) ; and for the principles 

 of the theory of complex numbers and complex quadratic forms, to Lojeune 

 Dirichlet's Memoir, " Recherches sur les formes quadratiques a coefficients 

 et a indeterminees complexes" (Crelle, vol. xxiv. p. 291). 



I. The Definition of the Genera. 

 Let f={a, 5, c) be an uneven* primitive form of determinant D, and 

 m=ax^-\- 2bxy + cy^ m' = ax'^ + 2bx'y' + cy'^ two numbers represented by /. 

 The generic characters of / are deducible from the equation 

 {ax^-^2hxy-\-cy'') {ax^-\-2bx'y' -{-cy") = 



{axx H- hlxy' + xy'\ + cyy'y —D^xy' —xyY, 



* A primitive form (a, b, c) is uneven, semieven, or even, according as the greatest 

 common divisor of a, 2b, c is 1, 1-f?, or {l+if; i.e., in Lejeune Dirichlet's nomencla- 

 ture, according as {a, b, c) is of the first, second, or third species. In this paper, when 

 we speak of an imeven, semieven, or even form or class, we shall always suppose the 

 form or class to be primitive. A semieven number is a number divisible by 1 but not 

 by(i+?7. 



